Math Assignment

Name Matthew Ramirez CHM100 Section 029LS

1. Solve the following: a) 256 × 0.05 =

b) 2.5 × 100 =

c) d) e)
2. Express the following exponentially: a) 10,000 c) 0.0015

b) 150 d) 0.023

3. Express the following as whole numbers:

a) c)

b) d)

4. If

a) Write an equation for c in terms of a and b

b) Write an equation for b in terms of a and c

5. Given that

a) Find

b) Find

6. Given the following data, fine the average

35.3 35.6 34.9 35.0

7. 2.5 centimeters = 1.00 inches

a) How many centimeters are there in 5.0 inches?

100

b) How many inches are there in 21.0 centimeters?

8. The density is defined as the mass of a substance divided by its volume. Determine the density of

9.0 mL of a liquid with a mass of 13.4 grams.

9. The density of mercury is 13.5 g/mL. Determine the number of grams found in 30.0 mL of mercury?

10. A sample weighing 30.0 grams contains 1.5 grams of salt. Determine the percent salt found in the sample.

11. A sample contains 25% water and weighs 201 grams. Determine the grams of water in thesample.

Experiment 1. Metric System and Measurements

OBJECTIVES:

(a) You will become familiar with the metric system of measurements.

(b) You will make measurements of mass, length, temperature and report you answer to the correct number of significant digits (significant figures). You will become familiar with rules for reporting the correct number of significant digits when adding, multiplying, dividing and subtracting measurements.

(c) You will become familiar with using measuring devices (balances) and different types of glassware.

INTRODUCTION

In chemistry, you will use two types of numbers:

a. Exact numbers and

b. Numbers that result from measurements

Exact numbers

An exact number is a number that arises when you count items or sometimes when you define a unit. For example, there are 20 students in a class. 20 is an exact number because there are exactly 20 students not 20.00001 or 19.9999. Exact numbers have an infinite number of significant digits. or

1 dozen = 12. Twelve in this definition is an exact number, because a dozen is exactly 12, not 11.99999 or 12.00000001 etc. but exactly 12.

Measured Numbers

A measurement is a quantity with a unit.

2 is just an exact number.

2 cm is a measurement of length. It has a number followed by a unit

In this experiment, you will make measurements using the metric system of units. The metric system is a system of units of measurement established from its beginnings in 1874 by diplomatic treaty, to the more modern General Conference on Weights and Measures – CGPM (Conferérence Générale des Poids et Measures). The modern system is actually called the International System of Units or SI. SI is abbreviated from the French Le Système International d’Unités and grew from the original metric system. Today, most people use the name metric and SI interchangeably, with SI being the more correct title.

SI or metric is the main system of measurement units used in science today. Each unit is considered to be dimensionally independent from each other. These dimensions are described as the measurements of length, mass, time, electric current, temperature, amount of a substance, and luminous intensity.

Base SI units used in chemistry

Physical quantity

Name of unit

Abbreviations

Length

Meter

m

Mass

Kilogram

kg

Time

Second

s

Temperature

Kelvin

K

Amount of substance

Mole

mol

Electric Current

Ampere

A

Luminous Intensity

Candela

cd

In the SI system, you have the base units of measurement, as shown in the table above. Base units in the Metric System can be converted into units that are more appropriate for the quantity being measured by adding a prefix to the name of the base unit. These prefixes are based on multiples of 10 and when placed before the base unit makes them larger or smaller. A millimeter is one thousandth of a meter and the symbol for a millimeter is mm and a milligram is one thousandth of a gram and the symbol is mg.

Greek Prefixes used with SI Units and Greek Prefix and symbol for prefix

Prefix

Meaning

Pico (p)

one trillionth (10-12)

Nano (n)

one billionth (10-9)

Micro ( )

one millionth (10-6)

Milli (m)

one thousandth (10-3)

Centi (cm )

one hundredth (10-2)

Deci (d)

m: meter

one tenth (10-1)

1 cm

10 cm

100 cm = 1 m

1000 mm = 1 m

1 Km = 1000m

one thousand (103)

Mega (M)

one million (106)

Giga (G)

one billion (109)

Derived SI Units

Derived units are associated with derived quantities, for example velocity is a quantity that is derived from the base quantities of time and length which, in SI, has the dimensions meters per second (symbol m/s or ms-1). The dimensions of derived units can be expressed in terms of the dimensions of the base units. In the table below are some examples of derived SI units.

Physical quantity

Name of unit

Abbreviation

Volume

cubic meter

m3

Pressure

Pascal

Pa

Energy

Joule

J

Electrical charge

Coulomb

C

UNCERTAINTY IN MEASUREMENTS

There is always an uncertainty in any measurement.

MEASURING VOLUME IN A GRADUATED CYLINDER

First, note that the surface of the liquid is curved. This is called the meniscus. This phenomenon is caused by the fact that water molecules are more attracted to glass than to each other (adhesive forces are stronger than cohesive forces). When we read the volume, we read it at the bottom of the meniscus.

26.5 ml 1.00000 cm3 = 1.000000000 ml

The smallest division of this graduated cylinder is 1 mL. Therefore, our reading error will be ± 0.1 mL. An appropriate reading of the volume is 26.5 ± 0.1 mL. An equally precise value would be 26.6 mL or

26.4 mL. We have three significant figures. The “5”, the last digit, is the uncertain figure.

Significant figures (significant digits) are those digits in a measured number (or result of a calculation with measured numbers) that include all certain digits plus a final one having same uncertainty.

Anytime you make a measurement, you must report the measurement to the correct number of

significant digits, i.e. the digits you are sure of, plus the last digit, which is estimated. When you make a measurement, you know and record the digit(s) you are sure of and a last digit, which is estimated.

However, if a measurement, made by other individuals, is reported to you, then you may be asked to determine the number of significant digits (significant figures). Significant figures are critical when you report scientific data because they give the reader an idea of how precisely you actually measured your data.

Number of significant figures refers to the number of digits reported for the value of a measured or calculated quantity, indicating the precision of the value. To count the number of significant figures in a given measured quantity, you observe the following rules.

1. All non-zero numbers in a measurement are always significant e.g. 2.563 cm – has 4 significant digits; 4.67546 g – has 6 significant digits, 4.8976569 mm has 8 significant figures (significant digits).

2. Zeros in a measurement may or may not be significant depending on their position within the measurement

(a) Zeros at the beginning of a measurement are never significant. Thus 7.34 cm, 0.734 cm and 0.00734 cm all contain three significant figures.

(b) Zeros between two non-zero digits are always significant e.g. 2.00005 cm has 6 significant digits; 4.03 kg has 3 significant digits, 0.008009 m has four significant digits.

(c) Terminal zeros ending at the right of the decimal point and to the right of a significant digitare significant. Each of the following has three significant digits (figures) 2.00 cm, 2.10 cm, 60.0

(d) cm, 0.00500 cm, 0.000707 mm.

3. Terminal zeros in which there is an implied decimal point are assumed to be not significant e.g., 70, 700, 7000 all have one significant digit. However, if you are explicitly told that some of the zeros are significant, they are significant. If the decimal point is explicitly shown zeros to the left of the decimal point and after a nonzero digit are significant e.g., 90. cm has 2 significant digits; 90.0 cm has 3 significant digits. However, 900 has one significant digit.

Note: Generally, rule 3 can be resolved by using scientific notation.

Example . How many significant figures are present in the following numbers?

Number

# Significant Figures

Rule(s)

40,123

5

1, 2b

4.567

4

1

800.00

5

1, 2b, 2c

0.0004

1

1, 2a

8.1000

5

1, 2c

500.040

6

1, 2b, 2c

3,000

1

3

10.0

3

1, 2b, 2c

RULES FOR DETERMINING THE NUMBER OF SIGNIFICANT DIGITS IN CALCULATED VALUES

i.e. significant digits and calculations

1. Multiplication and division

When multiplying, or dividing measured quantities, provide as many significant figures in the answer as there are in the measurement with the least number of significant figures.

2. Addition and subtraction

When adding, or subtracting measured quantities, provide the same number of decimal places in the answer as there are in the measurement with the least number of decimal places.

EXAMPLES USED IN THE RULES

NUMBER

Number of decimal places desired

Last figure to be kept (retained digit)

First figure to be dropped (dropped digit)

Last figure kept and/or number becomes

6.422

1

6.4

6.42

6.4

6.4872

2

6.48

6.487

6.49

6.997

2

6.99

6.997

7.00

6.6500

1

6.6

6.65

6.6

7.485

2

7.48

7.485

7.48

6.755000

2

6.75

6.755

6.76

8.995

2

8.99

8.995

9.00

6.6501

1

6.6

6.65

6.7

7.4852007

2

7.48

7.485

7.49

Example . Give the answer to the correct number of significant figures:

1.3332 cm + 1.345 cm + 0.89 cm =

Answer . When added on your calculator, the result that is shown is 3.5682. However, this has to be rounded to the lowest number of decimal places (two decimal places), since we are adding measurements. Hence the answer is 3.57 cm.

Example . Give the answer to the correct number of significant figures:

234 m ? 1.084 m =

Answer . 234 meters has 3 significant figures and 1.084 meters has 4 sig figs. In this calculation, the answer must have no more than 3 significant figures.

234 m ? 1.084 m = 253.656 m ?????Rounding 254 m2.

Example .

Now, it is time to combine the knowledge of “uncertainty” and significant figures. The following three examples will illustrate the importance of presenting a meaningful result after completing an experiment.

A metallic irregular object weighs 23.56 g. The piece is dropped in a 100-mL graduated cylinder, which contains 60.0 mL of water. The water level rises to 63.5 mL. What is the density of the metallic object?

Answer5. The volume of the object is 63.5 – 60.0 = 3.5 cm3 (Now this number has only 2 significant figures, not 3). The density (d) of the metallic object is

Density = Mass/Volume D = 8g / 4ml = 2 g/ 1ml = 2g/ 1cm3

d ?? 6.7 g /cm3

The result has only 2 significant figures. 1 cm3 = 1 ml = 1 cc

SCIENTIFIC NOTATION

In scientific notation, a number is written in the form A × 10n. A is a number greater than one and less than 10 and must have the same number of significant digits as were present in the original number. The exponent n is a positive or negative integer.

Question : Express the following in scientific notation:

a. 843.4

b. 0.00421

c. 845.00100

Answers:

a. 8.434 × 102

b. 4.21 × 10-3

c. 8.4500100 × 102

EXPERIMENTAL PROCEDURES

PART A

Measurement of Mass

1. Weigh a 100-mL beaker on a centigram balance and record the measurement in table “A”, below (remember a centigram balance measures to ± 0.01 g). Put a circle around the estimated digit in the measurement and determine the number of significant digits in the measurement. Note: A centigram balance measures to two decimal places.

2. Weigh the 100-mL beaker used in part “a” using a milligram balance and record the measurement in table “A” (remember a milligram balance reads to ± 0.001 g). Put a circle around the

estimated digit in the measurement and determine the number of significant digits in the measurement. Note: A milligram balance measures to three decimal places.

3. Repeat part “a” and “b” above, but this time using a 125 mL Erlenmeyer flask. Record your measurements in table A.

PART B
Determining the volume and the perimeter around face ABFE of the rectangular solid, shown in the diagram below.

Choose one of the vertices of the cube e.g. vertex B (Three edges meet to create a vertex). Measure the lengths of AB, BC and BF in centimeters, recording your measurements to the nearest +/- 0.01 cm in your data sheet.

Calculate the volume of the rectangular solid, showing your calculations in the space provided and reporting your answer to the correct number of significant digits.

Calculate the perimeter around face ABFE of the rectangular solid reporting your answer to the correct number of significant digits (perimeter means distance around face ABFE).

PART C
Calculating the density of a rectangular solid.

Your instructor will provide you with an unknown rectangular solid. Record the number or letter written on the rectangular solid. If your rectangular solid is a pure metal, record the symbol or name of the metal. Measure the length, width and thickness (height) of your rectangular solid in centimeters (note: your ruler reads to ± 0.01cm). Record your measurements in table B on your data sheets.

Weigh the rectangular solid on a centigram balance and record your mass to the nearest ± 0.01 g, in table B.

Calculate the density of your rectangular solid, reporting your answer to the correct number of significant digits and with the correct units.

Part D
Measuring volumes of liquids.

1. Fill a 100-mL measuring cylinder to the 100-mL mark with tap water and record its volume as

100.0 mL. Approximately, half fill a test tube with water from the cylinder and record the volume of the water left in the cylinder in table C. Remember a 100-mL measuring cylinder reads to ± 0.1 mL. Report the number of significant digits in each of the measurements you made, in Table C.

2. Select a 10-mL measuring cylinder that has 10 equal divisions per mL. Fill the 10-mL measuring cylinder to the 10-mL mark with tap water and record its volume as 10.00 mL. Approximately, quarter fill a test tube with water from the cylinder and record the volume of the water left in the cylinder in table C. Remember a 10-mL measuring cylinder reads to ±0.01 mL.

Report the number of significant digits in each of the measurements you made, in the Table C.

DATA SHEETS NAME:

PART A: Measurement of Mass

TABLE A

GLASSWARE

Mass

Number of significant digits

Mass of 100 mL beaker obtained using a centigram balance

Mass of 100 mL beaker obtained using a milligram balance

Mass of 125 mL Erlenmeyer flask obtained using a centigram balance

Mass of 125 mL Erlenmeyer flask obtained using a milligram balance

PART B
Determining the volume of a rectangular solid and perimeter around one of the faces (ABFE).

Choose one of the vertices of the cube e.g. vertex B (Three edges meet to create a vertex).

1. Length of AB in centimeters:

2. Length of BC in centimeters:

3. Length of BF in centimeters:

4. Volume of rectangular solid:

Show your calculation of the volume of the rectangular solid, in the space provided below, reporting your answer to the correct number of significant digits.

Perimeter around Face ABFE:

Show your calculation of the perimeter around face ABFE, in the space provided above, reporting your answer to the correct number of significant digits.

PART C
Calculating the density of a rectangular solid.

:

TABLE B

Length of rectangular solid

Width of rectangular solid

Height (thickness)of rectangular solid

Mass of rectangular solid

200.00 g

Calculation of Density of rectangular solid ; D = g/ml, =

Density of Rectangular =

PART D
Measuring volumes of liquids.

TABLE C

VOLUME MEASUREMENT

Volume (mL)

Number of significant digits

Volume of tap water in the P/measuring cylinder when it is filled to the 100-mL mark:

100.05

Volume of tap water left in the measuring cylinder after approximately half-filling a test tube:

50.00

Volume of tap water in the P/measuring cylinder when it is filled to the 10-mL mark:

10.0

Volume of tap water left in the measuring cylinder after approximately quarter-filling a test tube.

2.5

POSTLAB QUESTIONS NAME:

(Note that all the numbers are measured numbers)

1. How many significant digits are there in each of the following measurements?

(a) 2688 cm (b) 3.507 g

(c) 5.700 km (d) 0.00400 kg

(e) 24.00300 m (f) 2400 m

(g) 9.5000 x 10-8 m (h) 0.0020056 g

2. Round each of the following measurements to two significant digits.

(a) 0.00258 cm

(b) 28.23 g

(c) 5.998 cm

(d) 1.75 g

(e) 1.05 g

3. Do the following calculations and round your answer to the correct number of significant digits.

(a) 16.5 cm + 8 cm + 4.37 cm = (b) 13.57 cm – 6.3 cm =

(c) 672 cm × 39.864 cm × 10 cm = (e) 13. 57 cm / 6.3 cm2 =

(f) (14.86 cm + 13.7 cm) × (65.346 cm – 4.10 cm)

=

(43.888 cm – 32.888 cm)

4. Write each of the following measurements using scientific notation.

(a) 0.000450300 m

(b) 4567.67 g

5. Why are significant digits important when taking data in the laboratory?

6. What is wrong asking about the number of significant digits in 4.56?

7. Indicate how many significant figures there are in each of the following measured values.

a. 246.89

=

b. 107.856

=

c. 100.2

=

d. 0.646

=

e. 1.006

=

f. 0.005640

=

g. 14.600

=

h. 0.0002

=

i. 800000

=

j. 350.670

=

k. 1.0000

=

l. 320001

=

8. Calculate the answers to the appropriate number of significant figures.

a. 32.456 + 135.0 + 1.2345 =

b. 246.14 + 238.234 + 98.1 =

c. 658.0 + 23.34567 + 1345.29 =

9. Calculate the answers to the appropriate number of significant figures.

(a) 24.7 × 1.8 =

(b) 45.65 × 0.25 =

(c) 81.04 g × 0.010 =

(d) 6.47 × 63.7 =

3.

(e) ?

0.

(f) 1.678 ??

0.

(g) (2.67×1012) × (3.7 × 104) =

1.2X103

(h) 2.4X104

10. Round off the numbers to the indicated significant figures.

(a) 230 to one sig fig

(b) 2345 to three sig fig

(c) 789000 to two sig fig

(d) 6700 to one sig fig

11. Convert the following numbers to scientific notation using significant figures.

a. 2340000000

b. 0.00034

c. 123000000

d. 0.000160

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